Numbers from 0 to 1

Let's do an experiment. Instead of taking all the real numbers just take $0.9, 0.99, 0.999$ and so on, which are less than one. $\begin{aligned} S&=0.9+0.99+0.999+\cdots\\&=(1-0.1)+(1-0.01)+(1-0.001)+\cdots\end{aligned}$ Clearly, we can see that the above sum is tending to $\infty$ . So, the sum of all real numbers from $0$ to $1$ is $\boxed{\infty}$ .